Is there a mathematical formula to determine the length of material on
a roll, given the outside diameter of the core, the outside diameter
of the whole roll, and the thickness of the material (determined by a
micrometer)?

I was able to make a picture showing why (a+b)^2 = a^2 + 2ab + b^2 by
having a square with sides a+b and getting four areas within the
square. But I can't figure out how to show that a^2 - b^2 =
(a+b)(a-b) using a similar diagram.

Two circles, one of radius 5, the other of radius 8, intersect at
exactly one point, and the center of each circle lies outside the
other circle. A line is externally tangent to both circles. Find the
distance between the two points of tangency.

Draw the locus of all points in the plane that are equidistant from
the rays of an angle and equidistant from two points on one side of
the angle. I'm having trouble with this kind of locus problem. Can you
explain how to think about and solve them?

I have found a general formula to work out the side length a polygon (of
any shape) must have to be equable, which is closely linked to finding an
equation for the area of a circle without using pi (and therefore proving
pi)...

Can you help me show, with and without calculus, that the geometric figure of a maximum area and given perimeter is a circle? What are the dimensions of a triangle with perimeter p that encloses the maximum area?